On the investigation of free vibrations of nonthin cylindrical shells of variable thickness by the spline-collocation method

For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization. The problem has been solved within the framework of the refined Timoshenko–Mindlin theory. We have also investigated the influence of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions.     

弹性基上锥壳一般弯曲问题的精确解

本文应用Donnell的简化假定,从弹性基上锥壳位移型微分方程组出发,通过引入一个位移函数U(s,θ)(在极限情况下就退化成V.S.Vlasov对于圆柱壳所引的位移函数[5]),将基本微分方程组化成为一个八阶可解偏微分方程.这个方程的一般解用级数形式给出.对于在实际中有广泛应用价值的Winkler弹性基上锥壳的轴对称弯曲问题,本文给出了详细的数值结果,并求出了边缘荷载作用下的影响系数,这对计算弹性基上锥壳组合结构有着重要的意义.     

A study of the perturbed stress state of a closed cylindrical shell with a system of part-through transverse cracks

We study the elastic equilibrium of a closed infinite circular cylindrical shell with a system of surface cracks of identical length and depth. We use the method of singular integral equations together with the modeling of solid matter in the plane of a part-through crack by irregularly distributed “line springs”. We conduct a numerical analysis of the variation of the relative stress intensity factor at the center of a crack as a function of the parameters of a crack and the number of cracks. We study cracks located on both the interior and exterior surface of the shell. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 63–65.     

Inverse problem of nondestructive optical testing of residual stresses in cyclindrical shells

We considered the inverse problem of determination of the stress-strain state of a circular cylindrical shell that consists of two heterogeneous parts joined by a circular seam. As the input data for this problem, the results of polarization-optical measurements are used. The problem is formulated within the framework of a model that considers residual stress as caused by the geometric incompatibility of elastic deformations. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 151–156, April–June, 1998.     

Fluid dynamics of an underwater electric discharge on the axis of a shell immersed in a fluid

We study the motion of a fluid during an electrical discharge on the axis of an elastic cylindrical shell immersed in the fluid. We estimate the influence of the parameters of the shell on the dynamics of a gasvapor chamber. We point out several properties of the computation of the pulsed flows in an infinite volume of fluid (the choice of the method of constructing the grid, the limitation of the computational region). We obtain the pressure distribution over space at large distances from the shell. We analyze the influence of the shell on the parameters of the flow in the interior and exterior regions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 120–126.     

Analysis of contact stresses in structural joints of composite shell with steel banding

Based on the generalized Timoshenko-type shell theory, a numerical-analytical procedure for determining contact stresses from the interaction between a cylindrical composite shell and rigid bandings is proposed. Specific cases of loading and contact interaction (ideal contact through an adhesive interlayer) are considered. The contact problems are reduced to the solution of a Fredholm integral equation of the second-kind. A calculation analysis is performed. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 109–120, January–February, 2000.     

Thermoelastoplastic deformation of complexly reinforced shells

The problem on the elastoplastic deformation of reinforced shells of variable thickness under thermal and force loadings is formulated. A qualitative analysis of the problem is carried out and its linearization is indicated. Calculations of isotropic and metal composite cylindrical shells have shown that the load-carrying capacity of shell structures under elastoplastic deformations is several times (sometimes by an order of magnitude) higher than under purely elastic ones; the heating of shells with certain patterns of reinforcement sharply reduces their resistance to elastic deformations, but only slightly affects their resistance to elastoplastic ones; not always does the reinforcement in the directions of principal stresses and strains provide the greatest load-carrying capacity of a shell; there are reinforcement schemes that ensure practically the same resistance of shells at different types of their fastening. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 707–728, November–December, 2006.     

Postbuckling of a shear-deformable anisotropic laminated cylindrical shell under external pressure in thermal environments

A postbuckling analysis is presented for a shear-deformable anisotropic laminated cylindrical shell of finite length subjected to external pressure in thermal environments. The material properties are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear-deformation shell theory with the von Karman-Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The boundary-layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling region, and the initial geometric imperfections of the shell, is extended to the case of shear-deformable anisotropic laminated cylindrical shells under lateral or hydrostatic pressure in thermal environments. The singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The results obtained show that the variation in temperature, layer setting, and the geometric parameters of such shells have a significant influence on their buckling load and postbuckling behavior. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 789–822, November–December, 2007.     

Vibrations of a ribbed cylindrical shell with elastically fastened faces

We study the free vibrations of a reinforced cylindrical shell of revolution with elastically fastened faces. The computational model presented makes it possible to take account of various structural characteristics, including reinforcing ribs and attached rigid bodies. The solution is constructed using the linear theory of thin elastic shells and rods and the Ritz method. Three figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 114–119.     

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

The problem of a thin spherical linearly elastic shell perfectlybonded to an infinite linearly elastic medium is considered.A constant axisymmetric stress field is applied at infinityin the matrix, and the displacement and stress fields in theshell and matrix are evaluated by means of harmonic potentialfunctions. In order to examine the stability of this solution,the buckling problem of a shell which experiences this deformationis considered. Using Koiter's nonlinear shallow shell theory,restricting buckling patterns to those which are axisymmetricand using the Rayleigh–Ritz method by expanding the bucklingpatterns in an infinite series of Legendre functions, an eigenvalueproblem for the coefficients in the infinite series is determined.This system is truncated and solved numerically in order toanalyse the behaviour of the shell as it undergoes bucklingand to identify the critical buckling stress in two cases, namely,where the shell is hollow and the stress at infinity is eitheruniaxial or radial.     

Reflection of harmonic elastic waves from a convex cylindrical cavity of arbitrary section

We describe an approach to the solution of problems of reflection of harmonic elastic waves from a convex indrical cavity of arbitrary section, on the basis of Debye ray, series. We obtain a system of linear algebraic equations to determine the constants of integration and expressions for determining the stresses in the elastic medium surrounding the convex cylindrical cavity. We give the results of computation of the stresses that arise as a result of the action of a planar harmonic rarefaction wave on a cylindrical cavity in the shape of elliptic and parabolic cylinders and on a cylindrical cavity whose section is a Munger oval. Four figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 10–16, 1991.     

An experimental study of the vibrations of cylindrical shells with holes

We give the results of an experimental study of the frequencies and shapes of the natural vibrations of thin-walled cylindrical shells of revolution weakened by cuts; these results were obtained by the method of holographic interferometry. We establish the effect of the profile and the number of holes, reinforcement by elastic ribs, and attached solid bodies on the fundamental dynamic characteristics of the shell system. We describe the method by which the experiment was conducted. Four figures, 2 tables. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 121–125.     

Nonlinear buckling behavior of 3D-braided composite cylindrical shells subjected to internal pressure loads in thermal environments

The nonlinear buckling behavior of a 3D-braided composite cylindrical shell of finite length subjected to internal pressure in thermal environments is considered. According to a new micromacromechanical model, a 3D-braided composite may be treated as a cell system where the geometry of each cell strongly depends on its position in the cross section of the cylindrical shell. The material properties of the epoxy matrix are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear deformation theory of shells with a von Karman–Donnell-type kinematic nonlinearity and include thermal effects. The singular perturbation technique is employed to determine the buckling pressure and the postbuckling equilibrium paths of the shell.     

Nonlinear nonaxisymmetric deformation of composite shells of revolution

We discuss the results of the determination of the stress and displacement fields in nonaxisymmetrically loaded nonlinear-elastic shells of revolution. The original nonlinear system of equations is linearized in accordance with the method of variation of elastic parameters. The two-dimensional linear boundary-value problem is reduced to a sequence of one-dimensional problems, which are solved using a numerical method. We carry out an analysis of the stress-strain state of a conical shell made of a composite material of granular structure. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 80–83.     

Mathematical model of an internal crack in an elastoplastic cylindrical shell

We propose a mathematical model that makes it possible to reduce the problem of the stressed state and limit equilibrium of a cylindrical anisotropic elastoplastic shell with internal crack to a system of nonlinear singular integral equations with discontinuous functions on the right-hand sides. We construct an algorithm for numerical solution of such systems together with the conditions of plasticity and boundedness of stresses near the crack. For a transversally isotropic shell, we carry out a numerical analysis of the dependence of the opening of the internal crack front on the load and geometric and mechanical parameters. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 111–116, April–June, 1998     

锥壳的位移解及应用*

本文提出求锥壳方程通解的另一种方法——位移法.文中根据文献[1]给出的壳体基本关系,导出锥壳一般弯曲问题的位移方程组,然后通过引入一个位移函数U(s,θ)(在极限情况下,就变为对于圆柱壳所引入的位移函数),从而将锥壳基本方程组化成关于位移函数U(s,θ)的8阶可解偏微分方程(控制微分方程).对于一般弯曲问题,该方程的一般解以广义超几何函数给出;对于轴对称弯曲问题,用Bessel函数给出其一般解.作为锥壳位移解法的应用,讨论了Winkler地基模式上的锥壳的轴对称弯曲问题,给出数值结果.     

Structural damping in an equi-strength cylindrical shell with an elastic filler

We present a method for rational application of the deformation properties of a shell system with an elastic filler: design of a shell with variable thickness while preserving the load-bearing ability of the system as a whole. For the equi-strength shell thereby obtained we state and solve the mixed contact problem taking account of dry friction with nonmonotone loading, making it possible to estimate the structural hysteresis in the system. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 86–91.     

3-D thermo-elastic solution for continuously graded isotropic and fiber-reinforced cylindrical shells resting on two-parameter elastic foundations

This paper investigates the three-dimensional thermo-elastic deformation of cylindrical shells on two-parameter elastic foundations with continuously graded of volume fraction, subjected to thermal load. Suitable temperature and displacement functions that identically satisfy boundary conditions at the edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are solved by Generalized Differential Quadrature (GDQ) method. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded cylindrical shells that have a smooth variation of volume fractions through the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Effects of stiffness of the foundation and variations of different parameters of generalized power-law distribution on steady-state responses of the functionally graded cylindrical shell resting on elastic foundation are discussed. In addition, the effects of the FGM configuration are studied by considering the mechanical entities of different FGM fiber-reinforced cylindrical shells resting on elastic foundation. Some results are presented for the first time and some important conclusions are drawn.     

The influence of the stiffness of reinforcing elements on the deformed state of shells with a hole

We propose a method of experimental estimation of the influence of the stiffness of elements that reinforce the contour of a hole on the deformed state of shells under surface pressure. We obtain the experimental data for the deflections on the contour of a lateral hole in a cylindrical shell as a function of the stiffness characteristics and the caps. We compare the results of experiments with numerical computations and give an analysis of the theoretical and experimental data obtained. Three figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 67–71, 1991.     

Instability of thermoelastic interaction between a half-space and a rigid base through a thin liquid layer

We investigate the instability of thermoelastic interaction between elastic and rigid half-spaces through a liquid interlayer under the conditions of heat transfer across the interfaces. Due to the small thickness of the liquid layer, its influence on the temperature field is taken into account by the thermal resistance of the contact between the bodies, which depends on the normal displacement of the boundary of the elastic body. The pressure inside the liquid is equal to the external pressure applied to the bodies. We determined the critical value of the external heat flow for which the instability becomes possible in such a system and studied the dependence of this value on the parameters of the elastic half-space, the thickness of the liquid layer, and its thermal conduction. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 76–82, April–June, 1998.